Mathematical Modeling of Cryochemical Formation

of Medicinal Substances in Nanoforms.

The Role of Temperature

and Dimensional Parameters.

IRINA ASTASHOVA1,3∗, GREGORY CHECHKIN1, ALEXEY FILINOVSKIY1,4,

YURIY MOROZOV2,4, TATYANA SHABATINA2,4

1Faculty of Mechanics and Mathematics, Lomonosov Moscow State University,

Leninskie Gory, 1, 119991, Moscow,

RUSSIA,

2Faculty of Chemistry, Lomonosov Moscow State University,

Leninskie Gory, 1-3, 119991, Moscow,

RUSSIA,

3Plekhanov Russian University of Economics,

Stremyanny lane, 36, 117997, Moscow,

RUSSIA,

4Bauman Moscow State Technical University,

2nd Baumanskaya street, 5, 105005, Moscow,

RUSSIA

∗Corresponding Author

Abstract: The work is aimed at creating a mathematical model of cryochemical synthesis of nanoforms of phar-

maceutical substances. The therapeutic efficacy of pharmaceutical substances largely depends on the size and

morphology of the particles. Reducing the particle size of pharmaceutical substances to nanoscale makes it pos-

sible to obtain highly effective drugs, which makes it possible to use smaller doses of drugs and, thus, reduce

side effects and toxicity. Cryochemical synthesis is one of the most powerful methods for obtaining nanoforms

of medicament. The method, which is completely new, is based on sublimation or evaporation of the initial phar-

maceutical substance under high vacuum conditions and the introduction of the resulting vapors into an inert gas

stream, followed by low-temperature condensation of the flow of molecules of the substance from the gas phase

on the cooled surface. The first step in the mathematical modeling of cryochemical synthesis processes is the

calculation of the temperature field in the carrier gas flow interacting with the cooled surface. For this purpose,

a stationary equation of thermal conductivity with mass transfer is used for the one-dimensional case. We prove

existence and uniqueness theorems of the solution. Analytical solutions of the equation for Dirichlet, Neumann

and Robin boundary conditions are found.

Received: June 29, 2022. Revised: September 22, 2023. Accepted: October 6, 2023. Published: October 16, 2023.

1 Introduction

The dimension and morphology of medicinal sub-

stances particles effect the therapeutic efficiency of

different antibacterial, antiviral, anti-inflammatory,

anesthetic, analgesic, cardiological sedative and soo-

thing etc. medicaments, [1], [2].

These parameters can determine the qualities of

medications such as bioefficiency and bioavailability.

Lowering drug powders particles size down to nano

sizes allows us to obtain the medications with high

medical efficiency due to a huge total surface area

and permits to use lower medication doses and thus

Key-Words: Cryochemical Formation, Medicinal Substances, Nanoforms, Inert Gas Flow, Mathematical

Modeling, Boundary-Value Problem, Existence and Uniqueness.

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to decrease the side effects and toxicity, [3]. Such ap-

proach is based on the development of new methods

for obtaining of nanoforms of medical substances.

One of the powerful methods for production of

medications nanoforms is cryochemical modification

of drug substances. The method is based on the

sublimation or evaporation of drug substance under

high vacuum conditions and incorporation of the va-

pors obtained into the stream of inert gas followed by

the low temperature condensation of drug substance

molecular beam from gas phase on the cooled sup-

port surface. The cryochemical production of drug

nanoparticles with the definite size and morphology

and narrow size distribution led to the obtaining of

medications with desired properties and led to the cre-

ation of innovative cryochemical technology for drug

nanoforms production. The technological stages of

cryochemical modification process are following.

The initial drug substance was heated up to def-

inite temperature in the previously heated warm in-

ert gas flow. The molecular beam of drug sub-

stance vapor incorporated into the inert gas flow

and transferred to the vacuumed cryochemical rector

volume, supplied by cooled support. Reaching the

cooled surface of support drug substance molecular

beam rapidly cooled and high supersaturation of drug

molecular vapor in the gas phase was achieved. This

led to the conditions allowed rapid gas-phase nucle-

ation and formation of many nanocrystal germs - the

embryo of new nanophase of drug substance. Fur-

ther growth of nanocrystals obtained was limited by

rapid decreasing of drug substance molecules con-

centration in the gas phase near the cooled support

surface. These facts make it possible to obtain drug

particles of the size close to the dimension of crit-

ical crystal germ –about several dozen of nanome-

ters. The exact size of drug nanoparticles depends

on the ratio of the rates of nanocrystals nucleation

and nanocrystals growth. The behavior of these pro-

cesses is determined by several experimental parame-

ters: temperature of the initial drug substance evapo-

ration/sublimation, temperature of gas-phase, the gas-

carrier flow, the size and construction geometry of

the cryochemical reactor and the temperature of the

cooled support.

The goal of this work lies in the creation of mathe-

matical models describing the physico-chemical pro-

cesses occurring during a cryochemical modification

of drug substances and determination of the optimal

combination of technological (experimental) parame-

ters of these processes for two antibacterial drug sub-

stances, dioxidine and chloroamphenycol, as exam-

ples.

The work on mathematical description of cry-

ochemical modification of drug substances can be di-

vided in two main tasks:

- the description of the temperature field of the

molecular beam incorporated in the inert gas flow

and interacted with the cooled support surface;

- the creation of the kinetic model, a model that

takes into account the processes of drug nano-

phase nucleation and nanocrystals growth in the

gas phase in the given temperature field.

Nowadays the needs of modern medicine and

pharmacology to develop new drugs is determined by

the fact that well-known, long-established drugs do

not meet the requirements of the 21st century. How-

ever, the creation and testing of new molecular forms

of drugs (drug discovery) require not only huge ma-

terial costs, reaching several billion dollars, but also

long, reaching several years, time spent on various

clinical and preclinical tests, at the cost of which are

human lives.

From this point of view, another approach is more

promising to increase the effectiveness of known

medicines and improve methods of their targeted de-

livery (drug delivery). This can be achieved, for ex-

ample, by reducing the particle size of the active drug

up to the nanoscale state, [4], as well as by synthe-

sizing new or obtaining previously known thermody-

namically metastable polymorphic modifications in

the form of kinetically stable forms, [5].

Various physical and chemical methods are used

to produce nanoforms of drugs, [6], which are divided

into top-down and bottom-up methods, [7]–[9]. The

essence of top-down methods is to reduce the parti-

cle size by mechanical action on initially large parti-

cles of the source substance. In this direction, "dry"

and "wet" mechanical crushing is most often used in

special mills, [10], including using low temperatures

(cryo-milling), [11]. High-pressure homogenization

methods have also been developed, [12], techniques

using nanoporous membranes, [13], [14] and others.

On the contrary, the bottom-up approach consists in

converting the initial Pharmacopoeia drug into a ho-

mogeneous state, which is a collective of individual

molecules or small molecular associates, and creating

conditions for the subsequent Assembly of new phase

embryos, their growth and formation of nanoparticles,

[15]–[17]. Examples of such techniques are: solvent

replacement method, [18], synthesis using supercriti-

cal fluids, [19]–[21], techniques including freeze dry-

ing (spray drying), [22]–[24], cryochemical synthe-

sis, [25]–[27].

The size is of primary importance, since the size

parameters largely determine the bioavailability of the

drug, [28]. For example, reducing the particle size

of the antigonadotropic drug dibazole in an aqueous

suspension from 10 microns to 169 nm led to an in-

crease in absolute bioavailability from 5.1±1.9% to

82.3±10.1%, [29]. Increasing bioavailability allows

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to reduce the therapeutic dose and, consequently, pos-

sible side effects. The effect of particle size may

consist not only in changing the solubility and rate

of dissolution, but also in enabling the penetration of

drug particles through the body's biological barriers,

[30]–[31].

2 Problem Setting

Essentially, the technology of cryochemical modifica-

tion of pharmaceutical substances is as follows. The

initial substance is heated up to a certain temperature

in a stream of preheated carrier gas. The resulting va-

pors of the initial compound are captured by the car-

rier gas stream and carried through the nozzle into the

external free vacuum space. The scheme of the ex-

periment is shown in Figure 1 with

1 - heated copper screen (mixed molecular flow

shaper),

2 - cold carrier gas flow,

3 - copper screen heater,

4 - copper–constantan thermocouple junction for de-

termining and regulating the temperature of the

copper screen,

5 - initial substance in the container,

6 - nozzle of the mixed molecular flow shaper,

7 - heated flow a carrier gas carrying vapors of a

medicinal substance (mixed molecular stream),

8 - a cooled mixed molecular stream,

9 - a cold surface (when cooled with liquid nitrogen,

the surface temperature is about 77 K (−196◦C),

10 - the flow front of the carrier gas,

11 - a layer of cryochemically modified substance.

Figure 1: Scheme of the experiment.

When the mixed molecular stream moves from the

nozzle of the shaper to the cold surface, it is sharply

cooled by the mechanism of thermal conductivity. As

a result, the gas phase turns out to be multiple super-

saturated relative to the equilibrium pressure of satu-

rated vapors of the compound and conditions are be-

ing created in the system for rapid gas-phase nucle-

ation. In turn, the high rate of nucleation quickly im-

poverishes the gas phase with the vapors of the com-

pound, which limits the further growth of crystallites.

Thus, it is possible to obtain crystallites with sizes

close to the sizes of critical nuclei, which are several

tens of nanometers for organic compounds. The task

to describe mathematically the process of cryochem-

ical modification of pharmaceutical substances is di-

vided into two parts:

- calculation of the temperature field in the stream

of the carrier gas interacting with the cooled sur-

face;

- construction of a kinetic model that takes into ac-

count the processes of nucleation and growth of

nanoparticles in the gas phase in a given temper-

ature field.

To solve the first problem, we need to make a num-

ber of assumptions:

- the mixed molecular stream does not dissipate

when moving from the nozzle of the molecular

stream shaper to the cold surface,

- the mixed molecular stream has the same temper-

ature throughout the cross section,

- the temperature of the mixed molecular stream is

equal to the temperature the cooled surface when

reaching it,

- thermophysical characteristics of the carrier gas

do not change when the molecules and nanoparti-

cles of the pharmaceutical substance are included

in it.

Under these assumptions, the thermal conductivity

equation with mass transfer for the one–dimensional

case can be used to calculate the temperature field cre-

ated by the carrier gas stream:

∂T

∂t =V∂T

∂x −µ

ρCV·∂

∂x λ∂T

∂x .(1)

Here ρ, µ, λ are the density kg/m3, molecular

weight (kg/mol), thermal conductivity (W/(m·K))

of the carrier gas, respectively, CVis the molar

heat capacity of the carrier gas at constant volume

(J/(mol ·K)),Vis the linear velocity of the carrier-

gas flow front (m/s).

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In stationary mode we have ∂T /∂t = 0 and equa-

tion (1) reduces to the ordinary differential equation

dx −µ

ρV CV·d

dx λdT

dx = 0.(2)

The flow rate of the carrier gas is controlled during

the experiment with the help of an external device (an

industrial gas pipeline with accuracy, according to its

passport data, not worse than 5%). The regulated gas

stream of the carrier, passing through a heated copper

screen (a mixed molecular flow shaper) of cylindrical

shape, heats up to a certain temperature, captures the

vapors of the initial substance and takes them out into

the vacuum space. Let the nozzle area of the mixed

molecular flow shaper be S(m2)Then the molar flow

rate of the carrier gas is dN/dt(mol/s)and can be

written as

N=dN

dt =ρV S

µ.

In this case, the ratio of the molar flow rate of the

carrier gas dN/dt (mol/s)to the nozzle area of the

mixed molecular flow shaper, that is, the density of

the carrier gas flow dn/dt mol/(m2·s)can be rep-

resented as

˙n=dn

dt =˙

S=ρV

µ.

Now equation (2) can be written as

dx −d

dx λ

CV˙n·dT

dx = 0.(3)

It can be solved analytically, taking into account

the dependence of the thermal conductivity of the car-

rier gas on the temperature. An interesting fact is that

the heat capacity of gases in a wide range of pressures

practically does not depend on the pressure. This cir-

cumstance received its explanation from the molec-

ular kinetic theory. A large number of gases, such

as nitrogen, helium, argon, carbon dioxide, etc., have

the square-root dependence of the thermal conductiv-

ity on the temperature expressed by the approximate

formula

λ=ik

3π3/2d2RT

µ,(4)

where

iis the sum of translational and rotational degrees

of freedom of molecules (5 for diatomic gases, 3

for monatomic ones),

kis the Boltzmann constant,

µis the molar mass,

Tis the absolute temperature,

dis the effective diameter of molecules,

Ris the universal gas constant.

Representing λin (4) as α√Twith the appropriate

coefficient α, we obtain

CV˙n=α√T

CV˙n=b√Twith b=α

CV˙n.

Now equation (3) can be represented as

dx T−b√TdT

dx = 0, b > 0.(5)

3 General Decreasing Solutions

We study the dependence of the temperature on the

distance from the nozzle under three types of bound-

ary conditions, namely the Dirichlet, Neumann, and

Robin ones.

The Dirichlet condition specifies the temperature

value at the boundary.

The Neumann condition specifies the boundary

value for the derivative of the temperature.

In the Robin condition, we specify a linear combi-

nation of the temperature value and the derivative of

the temperature at the boundary.

The coefficient of the temperature value in the

Robin condition is the Biot number (the ratio of the

conductive thermal resistance inside the object to the

convective resistance at the surface of the object).

Note that similar problems were considered for a

heat process in [32].

Theorem 1. Each positive solution Tto equation (5)

is either constant or strictly monotonic. Each strictly

decreasing solution has the form

T(x) = c2Θx−x∗

bc 2

,(6)

where x∗and c > 0are arbitrary constants while Θ

is a decreasing function (−∞; 0) →(0; 1) implicitly

defined by

x= 2Θ(x) + ln 1−Θ(x)

1 + Θ(x).(7)

The left-hand side of (5) contains an expression

in parentheses, which must be constant and, for the

solution defined by (6), equals c2.

If maximally extended, such Tis defined on the

interval (−∞;x∗)and satisfies

T(x)→c2and T′(x)→0as x→ −∞,(8)

T(x)→0and T′(x)→ −∞ as x→x∗.(9)

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Proof. First, by the substitution T=Z2with Z > 0

we convert equation (5) into the form

Z2−2bZ2Z′′= 0,(10)

which immediately yields

Z2−2bZ2Z′=C=const

with further transformations depending on sgn C.

If C= 0, then either Z≡0or 1 = 2b Z′, which

entails that Z′>0and Zis strictly increasing.

If C=−c2<0, then we obtain Z2+c2=

2bZ2Z′. This shows again that Z′>0.

Finally, if C=c2>0with c > 0, then we obtain

Z2−c2= 2bZ2Z′.(11)

Now, if Z(x) = cat some point x, then, by the

uniqueness theorem, Zmust coincide with the con-

stant solution Z≡c. If not, then either Z > c on the

whole domain or Z < c. We reject the first case (with

Z′>0due to (11)) as well as the previous constant

one. In the second case we put

Z(x) = c zx

bc,0< z < 1,

which converts (11) into

z2−1 = 2z2z′.(12)

This can be written as

1 = 2z2z′

z2−1=2 + 2

z2−1z′,

whence, for 0< z < 1,

x−a=z(x)

02 + 2

ζ2−1dζ

=2z(x) + ln 1−z(x)

1 + z(x)

with some a. We have a general family of implicitly

defined strictly decreasing solutions to (12) satisfying

0< z < 1. One of them, with a= 0,is just Θ

defined by (7). All others can be obtained from Θby

a horizontal shift. Thus, we have (6).

It follows from (7) that

Θ(x)→0as x→0,

Θ(x)→1as x→ −∞.

Then, using (12), we obtain

Θ′(x)→ −∞ as x→0,

Θ′(x)→0as x→ −∞.

These limits, together with (6), produce the first three

limits in (8) and (9). For the fourth one, we use (11)

to obtain

T′= 2ZZ′=Z2−c2

2bZ =T−c2

2b√T→ −∞as T→0.

4 On Existence and Uniqueness of

Solutions

Theorem 2. For any constants x0< x1and T1>

T0>0, equation (5) has a unique solution Tdefined

on [x0;x1]and satisfying the conditions

T(x0) = T0, T (x1) = T1.(13)

Proof. The boundary conditions show that, according

to Theorem 1, the solution Tmust strictly decrease

and therefore have the form given by (6) and (7). So,

the boundary conditions become

Tj

c= Θxj−x∗

bc , j ∈ {0,1},

or, by using (7),

xj−x∗

bc = 2Tj

c+ln

1−Tj

1 + Tj

, j ∈ {0,1}.(14)

Thus, we have to prove the existence and uniqueness

of a pair (x∗, c)satisfying (14). Putting

q:= T1

T0∈(0; 1) and k:= √T0

c∈(0; 1),(15)

we write the difference of the two equations (14) as

k(x1−x0)

b√T0

= 2k(q−1) + ln (1 −qk)(1 + k)

(1 + qk)(1 −k)

or x1−x0

2b√T0

=Fq(k)(16)

with

Fq(k) := f(k)−qf (qk),(17)

f(k) := 1

2kln 1 + k

1−k−1.(18)

Lemma 3. For each A > 0and q∈(0; 1), there

exists a unique k∈(0; 1) such that Fq(k) = Awith

Fqdefined by (17) and (18). The mapping (A, q)7→ k

is a C1function (0; +∞)×(0; 1) →(0; 1) strictly

increasing with respect to both Aand q.

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Proof. Note that

f(k) = ln(1 + k)

2k−ln(1 −k)

2k−1,

whence f(k)→0as k→0(by L'Hôpital's rule) and

f(k)→+∞as k→1.

Now we study the derivative of fby using its Tay-

lor series uniformly converging on any subsegment of

the interval (0; 1).

f′(k) = 1

2k(1 + k)−ln(1 + k)

2k2

2k(1 −k)+ln(1 −k)

2k2

k(1 −k2)−ln(1 + k)

2k2+ln(1 −k)

2k2

∞



n=0

k2n+1

2k2

∞



n=1

((−1)n−1)kn

∞



n=0

k2n−1

∞



m=0

k2m+1

2m+ 1

∞



n=0 1−1

2n+ 1k2n=1

∞



n=1

2n+ 1k2n

∞



n=1

2n+ 1k2n−1>0,

whence f(k)>0as well.

Further, f′′ (k) =

∞



n=1

2n(2n−1)

2n+ 1 k2n−2>0,

whence f′is strictly increasing and

dFq

dk (k) = f′(k)−q2f′(qk)>0.

So, Fqis strictly increasing in k,Fq(k)→0as k→0,

and

Fq(k) = (1 −q)f(k) + qf(k)−f(qk)

>(1 −q)f(k)→+∞as k→1.

Therefore, Fqmust attain, exactly once, each A > 0,

which proves the first part of Lemma 3.

The second part follows immediately from the im-

plicit function theorem and the evident inequalities

∂(Fq(k)−A)

∂A =−1<0,

∂(Fq(k)−A)

∂q =−f(qk)−qkf′(qk)<0.

We return to proving Theorem 2. Having the

unique value of ksatisfying (16), we obtain, from (14)

and (15), the unique values

c=√T0

k>T0and

x∗=x1−2bT1−bc ln c−√T1

c+√T1

to satisfy (14). This completes the proof of Theo-

rem 2.

Now we are going to prove two theorems concern-

ing other boundary conditions for equation (5).

Theorem 4. For any real constants x0< x1,T0>0,

and U1<0, equation (5) has a unique solution T

defined on [x0;x1]and satisfying the conditions

T(x0) = T0, T ′(x1) = U1.(19)

Theorem 5. For any real constants x0< x1,T0>0,

and U1<0, equation (5) has a unique solution T

defined on [x0;x1]and satisfying the conditions

T(x0) = T0, T ′(x1) = U1T(x1).(20)

Proof. We try to prove the existence and uniqueness

of a constant T1∈(0; T0)such that the unique solu-

tion Texisting according to Theorem 2 satisfies the

boundary conditions of the related theorem.

According to Theorem 1, T−b√T T ′=c2,

whence, using notation (15),

T′(x1) = T(x1)−c2

bT(x1)=q2T0−T0/k2

bq√T0

=k2q2−1

k2q·√T0

T′(x1)

T(x1)=k2q2−1

k2q3·1

b√T0

where k∈(0; 1) is chosen, depending on q∈(0; 1),

to provide the boundary conditions (13) for the solu-

tion Tdefined by (6).

It follows from Lemma 3 that k∈(0; 1) strictly

increases with respect to q∈(0; 1). So, in both

right-hand sides of the last equations, the numerator

k2q2−1is negative and strictly increases in q, while

its absolute value decreases. The denominators are

positive and also strictly increase. Thus, the fractions

are negative with strictly decreasing absolute values.

Now consider their limits at 0 and 1.

Both fractions tend to −∞as q→0.As for q→1,

there must exist k1=lim

q→1k∈(0; 1].

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If k1<1, then it follows from (16)–(18) that

0<x1−x0

2b√T0

=F1(k1) = f(k1)−1·f(1 ·k1) = 0.

This contradiction shows that k1= 1. (For this k1, no

contradiction arises because f(k)→+∞as k→1.)

Hence

T′(x1)→0and T′(x1)

T(x1)→0as q→1.

So, both expressions strictly increase from −∞ to

0 as qincreases from 0 to 1 (i. e. as T1increases from

0 to T0). Therefore, they both must attain, exactly

once, each negative value, and this proves Theorems

4 and 5.

5 Illustrations

This section presents graphs of solutions to equation

(5) with various boundary conditions and various val-

ues of the constant b.

x, mm0 123456789

T, K

100

200

300

400

500

CH4

CO2

Figure 2: Solutions to a Dirichlet problem for various

gases.

Fig. 2 shows the solutions to (5) with the bound-

ary conditions T(0) = 473 K, T (0.01) = 77 K for

various values of the constant bcorresponding to var-

ious gases and the same flow rate ˙

N= 10 dm3/h (i. e.

1.156 ·10−4mol/s). Here bis equal to 0.002991 for

He, 0.00063456 for CH4, and 0.000374 for CO2(all

in m/√K).

Fig. 3 also shows dependence of solutions to (5)

on the constant brelated to various values of the flow

rate ˙

Nand the same gas CO2. Here the value of b

changes from 0.000374 (the topmost graph) to 0.0374

(the lowest one) m/√K, which correspond to the mo-

lar flow rates 1.156 ·10−4and 1.156 ·10−6mol/s.

Boundary conditions are the same as in Fig. 2.

Fig. 4 shows the solutions to (5) with the boundary

conditions T(0) = 473 K, T ′(0.01) = −105K/m

for various values of ˙

Nand the same gas N2with b

changing from 0.000393 to 0.0393 m/√K.

x, mm0 123456789

T, K

100

200

300

400

500

0.1

CO2

Figure 3: Solutions to a Dirichlet problem for various

flow rates ˙

x, mm0 123456789

T, K

100

200

300

400

500

0.1

Figure 4: Solutions with a Neumann right-end bound-

ary condition for various ˙

Fig. 5 shows the solutions to equation (5) satis-

fying the boundary conditions T(0) = 473 K and

T′(0.01)/T(0.01) = −103m−1for the same gas CO

and various values of ˙

N. Here the constant bchanges

from 0.0004169 to 0.04169 m/√K.

Fig. 6 shows the solutions to equation (5) sat-

isfying the boundary conditions T(0) = 473 K,

T′(0.01) = (−2,−4,−8) ·104K/m for the same

gas CH4and ˙

N= 10 dm3/h. Thus, here we have

b= 0.00063456 m/√K.

Fig. 7 shows the solutions to equation (5) satis-

fying the boundary conditions T(0) = 473 K and

T′(0.01)/T(0.01) equal to −102,−103,−104m−1

for the same gas Ar, the flow rate ˙

N= 10 dm3/h,

and therefore the same b= 0.0004506 m/√K.

6 Conslusions

We have analysed dependence of the temperature pro-

file on the distance from the cooled surface and com-

pared the results for various carrier gases such as

nitrogen, helium, argon, methane, carbon monoxide

and dioxide.

The above calculations show that

1) an increase in the thermal conductivity of the car-

rier gas, other parameters being constant, leads

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2023.20.22

Irina Astashova, Gregory Chechkin,

Alexey Filinovskiy, Yuriy Morozov, Tatyana Shabatina

E-ISSN: 2224-2902

219

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x, mm0 123456789

T, K

100

200

300

400

500

0.1

Figure 5: Solutions with a Robin right-end boundary

condition for various ˙

x, mm0 123456789

T, K

100

200

300

400

500

CH4

-80000

-40000

-20000

T’1

Figure 6: Solutions to Neumann problems with vari-

ous T′

to more uniform temperature distribution in the

stationary temperature field formed in the space

between the nozzle of the mixed molecular flow

and the cooled surface;

2) an increase in the flow rate of the carrier gas,

other parameters being constant, leads to an in-

crease of uneven temperature distribution in the

formed stationary temperature field.

Taking into account the identified patterns will

contribute, in the future, to the choice of optimal

regimes for cryochemical synthesis of nanoforms

of pharmaceuticals substances with specified dimen-

sional characteristics.

This is the first step in a complex investigation of

the role of various parameters in cryochemical syn-

thesis of various drugs.

The results obtained are confirmed by an experi-

ment on the formation of nanoforms for the antibacte-

rial drug dioxidine with the nitrogen as an inert carrier

gas.

Remark 1.The authors' results connected with math-

ematical modeling in other physical processes can be

found in [33]–[36].

x, mm0 123456789

T, K

100

200

300

400

500

-10000

-1000

-100

T / T1

’

Figure 7: Solutions to Robin problems with various

T′

1/T1.

Acknowledgment

This work was done with the support of MSU Pro-

gram of Development, Project No 23-SCH05-26.

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This work was done with the support of MSU Pro-

gram of Development, Project No 23-SCH05-26.

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DOI: 10.37394/23208.2023.20.22

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Alexey Filinovskiy, Yuriy Morozov, Tatyana Shabatina

E-ISSN: 2224-2902

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